direct product, metabelian, supersoluble, monomial
Aliases: C2×C3⋊D36, C6⋊2D36, D18⋊5D6, Dic3⋊4D18, C62.64D6, (C3×C18)⋊1D4, C3⋊3(C2×D36), C18⋊1(C3⋊D4), (C2×Dic3)⋊4D9, (C2×C6).18D18, (C3×C6).36D12, (C2×C18).19D6, (C22×D9)⋊2S3, (C6×D9)⋊5C22, (Dic3×C18)⋊5C2, C22.14(S3×D9), C6.19(C22×D9), C32.3(C2×D12), C6.7(C3⋊D12), (C3×C18).19C23, (C6×C18).13C22, C18.19(C22×S3), (C6×Dic3).10S3, (C3×Dic3).33D6, (C9×Dic3)⋊4C22, (C2×C6×D9)⋊1C2, (C3×C9)⋊4(C2×D4), C6.38(C2×S32), (C2×C6).25S32, C9⋊1(C2×C3⋊D4), C2.20(C2×S3×D9), (C22×C9⋊S3)⋊1C2, (C2×C9⋊S3)⋊5C22, C3.1(C2×C3⋊D12), (C3×C6).87(C22×S3), SmallGroup(432,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊D36
G = < a,b,c,d | a2=b3=c36=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1468 in 194 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C3×C9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, S3×C6, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, C3×D9, C9⋊S3, C3×C18, C3×C18, D36, C2×C36, C22×D9, C22×D9, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C9×Dic3, C6×D9, C6×D9, C2×C9⋊S3, C2×C9⋊S3, C6×C18, C2×D36, C2×C3⋊D12, C3⋊D36, Dic3×C18, C2×C6×D9, C22×C9⋊S3, C2×C3⋊D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C3⋊D4, C22×S3, D18, S32, C2×D12, C2×C3⋊D4, D36, C22×D9, C3⋊D12, C2×S32, S3×D9, C2×D36, C2×C3⋊D12, C3⋊D36, C2×S3×D9, C2×C3⋊D36
►On 72 points
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)
(1 25 13)(2 14 26)(3 27 15)(4 16 28)(5 29 17)(6 18 30)(7 31 19)(8 20 32)(9 33 21)(10 22 34)(11 35 23)(12 24 36)(37 49 61)(38 62 50)(39 51 63)(40 64 52)(41 53 65)(42 66 54)(43 55 67)(44 68 56)(45 57 69)(46 70 58)(47 59 71)(48 72 60)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(64 72)(65 71)(66 70)(67 69)
G:=sub<Sym(72)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36)(37,49,61)(38,62,50)(39,51,63)(40,64,52)(41,53,65)(42,66,54)(43,55,67)(44,68,56)(45,57,69)(46,70,58)(47,59,71)(48,72,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36)(37,49,61)(38,62,50)(39,51,63)(40,64,52)(41,53,65)(42,66,54)(43,55,67)(44,68,56)(45,57,69)(46,70,58)(47,59,71)(48,72,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63)], [(1,25,13),(2,14,26),(3,27,15),(4,16,28),(5,29,17),(6,18,30),(7,31,19),(8,20,32),(9,33,21),(10,22,34),(11,35,23),(12,24,36),(37,49,61),(38,62,50),(39,51,63),(40,64,52),(41,53,65),(42,66,54),(43,55,67),(44,68,56),(45,57,69),(46,70,58),(47,59,71),(48,72,60)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,9),(2,8),(3,7),(4,6),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(64,72),(65,71),(66,70),(67,69)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 54 | 54 | 2 | 2 | 4 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D9 | C3⋊D4 | D12 | D18 | D18 | D36 | S32 | C3⋊D12 | C2×S32 | S3×D9 | C3⋊D36 | C2×S3×D9 |
| kernel | C2×C3⋊D36 | C3⋊D36 | Dic3×C18 | C2×C6×D9 | C22×C9⋊S3 | C22×D9 | C6×Dic3 | C3×C18 | D18 | C2×C18 | C3×Dic3 | C62 | C2×Dic3 | C18 | C3×C6 | Dic3 | C2×C6 | C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 1 | 2 | 1 | 3 | 6 | 3 |
Matrix representation of C2×C3⋊D36 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 22 | 34 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 26 |
| 0 | 0 | 0 | 0 | 11 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 11 |
| 0 | 0 | 0 | 0 | 17 | 6 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,26,0,0,0,0,34,15,0,0,0,0,0,0,36,1,0,0,0,0,0,1,0,0,0,0,0,0,6,11,0,0,0,0,26,17],[1,27,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,31,17,0,0,0,0,11,6] >;
C2×C3⋊D36 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_{36} % in TeX
G:=Group("C2xC3:D36"); // GroupNames label
G:=SmallGroup(432,307);
// by ID
G=gap.SmallGroup(432,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^36=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations